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1 Classifying Cellular Automata Automatically: Finding Gliders, Filtering, and Relating Space-Time Patterns, Attractor Basins, and the Z Parameter ANDREW WUENSCHE Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico USA, wuensch Received January 2, 1998; accepted August 5, 1998 Cellular automata (CA) rules can be classified automatically for a spectrum of ordered, complex, and chaotic dynamics by a measure of the variance of input-entropy over time. Rules that support interacting gliders and related complex dynamics can be identified, giving an unlimited source for further study. The distribution of rule classes in rule-space can be shown. A byproduct of the method allows the automatic filtering of CA space-time patterns to show up gliders and related emergent configurations more clearly. The classification seems to correspond to our subjective judgment of space-time dynamics. There are also approximate correlations with global measures on convergence in attractor basins, characterized by the distribution of in-degree sizes in their branching structure, and to the rule parameter, Z. Based on computer experiments using the software Discrete Dynamics Lab (DDLab), this article explains the methods and presents results for 1D CA John Wiley & Sons, Inc. Key Words: attractor basins, cellular automata, filtering, glider dynamics, Z parameter 1. INTRODUCTION Cellular automata (CA) are a much-studied class of discrete dynamical network that supports emergent behavior resulting from homogeneous, local, shortrange interactions. They are applied in many overlapping areas: to model processes in physical, chemical, and biological systems such as fluid dynamics and reactiondiffusion [1,2]; to study self-organization and selfreproduction by the emergence of coherent interacting structures [3,4]; in mathematics and computation where the systems themselves are the focus of interest [5 7]. CA dynamics are driven by complex feedback webs that are difficult to treat analytically, except for special cases. Understanding these systems depends to a large extent on computer experiments, where a key notion is that state-space is connected into basins of attraction [8]. The ability of CA to support the emergence of coherent interacting long-lived configurations provides a striking ex John Wiley & Sons, Inc., Vol. 4, No. 3 COMPLEXITY 47 CCC /99/

2 ample of self-organization in a simple system and has consequently become the focus of particular study. This sort of behavior is characterized by interacting gliders (after Conway s 2D game-of-life [9]). Glider dynamics can be seen from a number of overlapping perspectives: Wolfram s complex (or class 4) behavior [10]; phase transitions between order and chaos [11]; computation [10,12]; and discrete analogues of Prigogine s far-from-equilibrium dissipative structures [13]. Because glider dynamics are relatively rare in CA rulespaces, their study has relied on the few known complex rules in 1D CA. A more general theory would benefit from a great many examples. Methods are described to classify rule-space automatically, for a spectrum of ordered, complex and chaotic dynamics, by a measure of the variance of input-entropy over time. This allows screening out CA rules that support glider (and related) dynamics, giving an unlimited source for further study. The resulting classification seems to correspond to our subjective judgment of spacetime dynamics. The method also gives statistical data on the distribution of rule classes in rule-space, for varying neighborhood sizes. Another useful byproduct allows automatic filtering of the space-time patterns of any CA to show up gliders and related emergent configurations more clearly. The quality of dynamical behavior of CA, from ordered to chaotic, 1 is approximately reflected by convergence in basins of attraction and subtrees (referred to collectively as attractor basins), in terms of their characteristic in-degree, which influences the length of transients and attractor cycles. The in-degree of a state is its number of pre-images (predecessors). Bushy subtrees with high in-degree imply high convergence and order. Sparsely branching subtrees imply low convergence and chaos. A simple convergence measure is G-density, the density of garden-of-eden states (those without pre-images) and the rate of increase of G-density with system size. For order these measures are relatively high, for chaos relatively low. A more general measure is the distribution of in-degree sizes. Generating attractor basins and making these measures relies on a reverse algorithm that computes preimages directly, without exhaustive testing. A consequence of the reverse algorithm is the rule parameter, Z, which predicts convergence by giving the probability that the next unknown cell in a partial pre-image is uniquely determined. This probability is calculated from the rule s look-up table [8]. This article defines the class of 1D CA in question, rules, trajectories, and basins of attraction, how these are represented, and the methods for computing preimages and the Z parameter. The characteristics of gliders and the methods for filtering space-time patterns and for automatically classifying rule-space are described, and results presented of the classified rule samples. Preliminary results are presented relating local measures (on trajectories), global measures (on attractor basins), and the Z parameter. The reasons why correlations are to be expected are discussed. The work is based on computer experiments using the author s software Discrete Dynamics Lab (DDLab) [14]. 2. 1D CA A CA is a regular network of elements (cells), taking inputs from their nearest (and next nearest and so on) neighbors according to a fixed neighborhood template, which defines the network geometry and the periodic boundary conditions. Cells synchronously update their cell-state according FIGURE 1 The space-time pattern of a 1D complex CA with interacting gliders. 308 time-steps from a random initial state. System size n = 700, neighborhood size k = 7, rule (hex) = 3b 46 9c 0e e4 f7 fa 96 f9 3b 4d 32 b0 9e d0 e0. Cells are colored/shaded according to neighborhood look-up instead of the value. Space is across and time down the page. The basin of attraction field for this rule for n = 16 is shown in Figure COMPLEXITY 1999 John Wiley & Sons, Inc.

3 to a homogeneous logical function on their inputs. The cellstate ranges over a discrete alphabet (in this article just a binary alphabet (0 or 1) is considered and the number of cells is finite, with periodic boundary conditions. Figure 2 shows neighbourhood templates for 1D CA as applied in DDLab (see [15] for 2D and 3D templates). A CA neighborhood of size k has 2 k permutations of values. The most general expression of the Boolean function or rule is a look-up table (the rule-table) with 2 k entries, giving 2 2k possible rules. Subcategories of rules can also be expressed as simple algorithms, concise AND/OR/NOT logical statements, totalistic rules [16], or threshold functions. By convention [16] the rule-table is arranged in descending order of the values of neighborhoods, and the resulting bit string converts to the decimal or hexadecimal rule number. For example, the k = 3 rule-table for rule 30, neighborhoods, decimal neighborhoods, binary outputs, the rule table The rule-table for other k values are set out in a corresponding way. k 4 rules are referred to by their hexadecimal rule numbers. k 3 rules are usually referred to by their more familiar decimal rule numbers. The behavior space of CA depends on the size of rule-space, 2 2k, though rule symmetries effectively reduce this number. For example, the 2 23 = 256 rules in k = 3 rule-space reduce to 88 equivalence classes [8]. 3. TRAJECTORIES AND SPACE-TIME PATTERNS A state of a CA is the pattern of 0s and 1s at a given timestep. A trajectory is the sequence of states at successive time-steps, the system s local dynamics. Examples are shown in Figures 1, 3, 4, and elsewhere. As well as showing cells as light (0) or dark (1), an alternative presentation shows cells in colors (or shades) according to their look-up neighborhood (Figure 3). The most frequently occurring colors can be progressively filtered to show up gliders and other space-time structures as in Figure 4, done interactively, on-the-fly, in DDLab for any CA. This is an alternative method to the computational mechanics approach [12]. 4. BASINS OF ATTRACTION The idea of basins of attraction in discrete dynamical networks (which includes CA) is summarized in Figure 5. Given an invariant network architecture and the absence of noise, a CA is deterministic and follows a unique trajectory from any initial state. When a state that occurred previously is revisited, which must happen in a finite statespace, the dynamics become trapped in a perpetual cycle of repetitions defining the attractor (state cycle) and its period (minimum one, a stable point). The approach is analogous to Poincaré s phase portrait in continuous dynamics. These systems are dissipative. A state may have multiple pre-images (predecessors) or none, but just one successor. The number of pre-images is the state s in-degree. In-degrees greater than one require that transient states exist outside the attractor. Tracing connections backward to successive pre-images of transient states reveals a tree-like topology where the leaves are states without pre-images, known as garden-of-eden states. Conversely, the flow in state-space is convergent. The set of transient trees and the attractor on which they are rooted make up the basin of attraction. Local dynamics connect state-space into a number of basins, the basin of attraction field, representing the system s global dynamics. An example is shown in Figures 6 and CONSTRUCTING AND PORTRAYING ATTRACTOR BASINS To construct a basin of attraction containing a particular state, the network is iterated forward from the state until a repeat is found and the attractor identified. The transient tree (if it exists) rooted on each attractor state is constructed in turn. Using the reverse algorithms, the pre-images of the attractor state are computed, ignoring the pre-image lying on the attractor itself, then the pre-images of pre-images, until all garden-of-eden states have been reached. FIGURE 2 The extra asymmetric cell in even k is on the right. The wiring is shown between two time-steps. 1D neighbourhood templates defined in DDLab, k = Another common notation defines the radius r of a symmetric neighborhood, r =(k 1)/ John Wiley & Sons, Inc. COMPLEXITY 49

4 FIGURE 3 Space-time patterns of a CA (n = 24, k = 3, rule 90). Twenty-four time-steps from an initial state with a single central 1. Two alternative presentations are shown. Left, cells by value, light = 0 dark = 1. Right, cells colored (or shaded) according to their look-up neighborhood. Just a subtree may be constructed rooted on a state. Because a state chosen at random is very likely to be a garden-of-eden state, it is usually necessary to run the network forward by at least one time-step and use the state reached as the subtree root. Running forward by more steps will reach a state deeper in the subtree, so allowing a larger subtree to be constructed. A considerable speedup in computation is achieved by taking advantage of equivalent dynamics because of rotated states and rotational symmetry [8], a property of the regularity of CA and periodic boundary conditions, resulting in equivalent subtrees and basins. Attractor basins are portrayed as state transition graphs, vertices (nodes) connected by directed edges. States are represented by nodes, by a bit pattern in 1D or 2D (as in Figure 7), or as the decimal or hex value of the state. In the graphic convention [8,14], the length of edges decreases with distance away from the attractor, and the diameter of the attractor cycle approaches an upper limit with increasing period. The direction of edges (i.e., time) is inward from garden-of-eden states to the attractor and then clockwise around the attractor cycle, as shown in Figure 7. Typically, the vast majority of states in a basin of attraction lie on transient trees outside the attractor, and the vast majority of these states are garden-of-eden states. 6. COMPUTING PRE-IMAGES CA attractor basins are constructed with an algorithm that directly computes the pre-images of network states [8,17]. The network is run backward in time, though backward trajectories usually diverge. The reverse algorithm takes advantage of the regularity of connections in 1D CA. It also works for mixed-rule networks. Other reverse algorithms 2 and methods designed for more general discrete dynamical networks can also be applied to CA. Provided k << n, the CA reverse algorithm is, in general, orders of magnitude faster FIGURE 4 Space-time patterns from the same initial state showing interacting gliders, which are embedded in a complicated background. Left, cells by value. Right, cells by neighborhood look-up, with the background filtered. The k = 3 rule 54 was transformed [8] to its equivalent k = 5 rule (hex) 0f3c0f3c, n = COMPLEXITY 1999 John Wiley & Sons, Inc.

5 FIGURE 5 For a network size n, an example of one of its states B might be State-space is made up of all 2 n states, the space of all possible bit strings or patterns. Part of a trajectory in state-space, where C is a successor of B, and A is a pre-image of B, according to the dynamics of the network. The state B may have other pre-images besides A; the total number is the in-degree. The pre-image states may have their own pre-images or none. States without pre-images are known as garden-of-eden states. Any trajectory must sooner or later encounter a state that occurred previously it has entered an attractor cycle. The trajectory leading to the attractor is a transient. The period of the attractor is the number of states in its cycle, which may be only just one a point attractor. Take a state on the attractor; find its pre-images (excluding the pre-image on the attractor). Now find the pre-images of each pre-image and so on, until all garden-of-eden states are reached. The graph of linked states is a transient tree rooted on the attractor state. Part of the transient tree is a subtree defined by its root. Construct each transient tree (if any) from each attractor state. The complete graph is the basin of attraction. Some basins of attraction have no transient trees, just the bare attractor. Now find every attractor cycle in state-space and construct its basin of attraction. This is the basin of attraction field containing all 2 n states in state-space but now linked according to the dynamics of the network. Each discrete dynamical network imposes a particular basin of attraction field on state-space. State-space and basins of attraction than the brute force method, constructing an exhaustive map resulting from network dynamics [15,17]. Some basic information on attractor basin structure can be found by statistical methods, first applied by Walker [18], as shown in Figure 8, and are appropriate for large networks. Trajectories are run forward from many random initial states looking for a repeat in the network pattern to identify the range of attractor types reached. The frequency of reaching a given attractor type indicates the relative size of the basin of attraction, and other data, such as the num John Wiley & Sons, Inc. COMPLEXITY 51

6 FIGURE 6 The basin of attraction field of the complex CA rule in Figure 1. n = 16, k = 7. The 2 16 = states in state-space are connected into 89 basins of attraction. The 11 nonequivalent basins are shown, with symmetries characteristic of CA [8]. The period (p), percentage of state-space in each basin type (s), and number of each type (t) of the biggest three basins (top row) are as follows: (1) p =1,s = 15.7%, t =1;(2)p =5,s = 55.8%, t = 16; (3) p = 192, s = 22.9%, t = 1. The field s G-density = 0.451, ratio = 0.938, Z = ber of basins, and the length of transients and attractor cycles, are extracted. These various methods are implemented in DDLab The CA reverse algorithm Consider a 1D CA size n (indexed n ) and neighborhood k. To find all pre-images of a state A, let P be a partial pre-image where at least k 1 bits (on the left), up to and including P i, are known. Now find the next unknown bit to the right, P i 1, consistent with the rule-table. ( indicates known, unknown, bits), P i+1 P i P i 1... partial pre-image P... compare the outputs of P i +1, P i, with each other and... known state A... A i with A i If k = 3 (for example), the bit string P i+1, P i, corresponds to two neighborhood entries in the rule-table. When their outputs, T 1 and T 2, are compared with each other and with A i, there are three possible consequences. The permutation is either deterministic, ambiguous, or forbidden. 1. Deterministic: If T 1 T 2, then P i 1 is uniquely determined, as there is only one valid neighborhood with the output A i. 2. Ambiguous: If T 1 = T 2 = A i, then both 0 and 1 are valid solutions for P i 1. The partial pre-image must be duplicated, with P i 1 = 0 in one version and P i 1 =1inthe other. 3. Forbidden: If (T 1 = T 2 ) A i, then P i 1 has no valid solution. If forbidden (3), the partial pre-image P is rejected. If deterministic or ambiguous (1 or 2), the procedure is continued to find the next unknown bit to the right. However, in the ambiguous case (2), both alternative partial preimages must be continued. In practice, one is assigned to a stack of partial pre-images to be continued at a later stage. As the procedure is reapplied to determine each successive unknown bit toward the right, each incidence of ambiguous permutations will require another partial pre-image to be added to the stack, though various refinements limit its growth. 52 COMPLEXITY 1999 John Wiley & Sons, Inc.

7 FIGURE 7 A detail of the second basin of attraction in Figure 6. The states are shown as 4 4bitpatterns. FIGURE 8 The attractor frequency histogram for the complex k = 5 rule 6c1e53a8, n = 50. The rule also appears in Figures 13 and 23. Max attractor period = 150, min = 4, max average transient = 681, min=6. The procedure is continued to the right to overlap the assumed start string, to check if periodic boundary conditions are satisfied; if so, the pre-image is valid. The procedure is reapplied to each partial pre-image taken from the partial pre-image stack, starting at the first unknown cell. Each time an ambiguous permutation (2) occurs, a new partial pre-image must be added to the stack, but the stack will eventually be exhausted, at which point all the valid preimages containing the assumed start string will have been found. The procedure is applied for 2 k 1 start strings, assuming the different possible values of the first k 1 bits. The reverse algorithm is applied from left to right in DDLab but is equally valid when applied from right to left The Z parameter Abyproduct of the CA reverse algorithm is the probability of the next unknown bit being deterministic (section 6.1(1)). Two versions of this probability are calculated from the rule-table: Z left for the reverse algorithm applied from left to right and Z right for the converse. The Z parameter is the greater of these values. For Z =1,itcanbe shown [8] that for any system size n, the maximum indegree, I max 2 k 1 because the next unknown bit is always uniquely determined, so the assumed start string of length k 1 may generate at most 2 k 1 pre-images. If only one of Z left or Z right =1,I max <2 k 1 because at least one assumed start string must be forbidden (section 6.2(3)). At the other extreme, for Z = 0, all state-space converges on the state all-0s or all-1s in one step. For high Z, low in-degree (relative to system size n) is expected in attractor basins, growing at a slow rate with respect to n. Conversely, for low Z, high relative in-degree is expected growing quickly with respect to n. High Z predicts low convergence and chaos; low Z predicts high convergence and order. The 2 k neighborhoods of size k, each indexed k , each has an ouput T (0 or 1) that makes up the rule-table (section 2) and may be expressed as a k 1, a k 2,...a 1, a 0 T. To calculate Z left from the rule-table, let n k be the count 1999 John Wiley & Sons, Inc. COMPLEXITY 53

8 of rule-table entries belonging to deterministic pairs such that, a k 1, a k 2,...a 1,0 T and a k 1, a k 2,...a 1,1 T (not T) The probability that the next bit is determined because of the above is given by R k = n k /2 k. This is a first approximation of Z left. Let n k 1 be the count of rule-table entries belonging to deterministic 4-tuples (where may be 0 or 1), such that a k 1, a k 2,...a 2,0, T and a k 1, a k 2,...a 2,1, T The probability that the next bit is determined because of the above is given by R k 1 = n k 1 /2 k. This count is repeated if necessary for deterministic 8-tuples where R k 2 = n k 2 / 2 k, 16-tuples, 32-tuples,... up to the special case of just one 2 k -tuple that occupies the whole rule-table. These are independent nonexclusive probabilities that the next bit is determined. The union of the probabilities R k R k 1 R k 2... = Z left is given by the following expression (the order of the probabilities makes no difference to the result): Z left =R k + R k 1 (1 R k )+R k 2 (1 (R k + R k 1 (1 R k ))) + R k 3 (1 (R k 2 (1 (R k + R k 1 (1 R k ))))) +... which simplifies to, Z left = R k + R k 1 (1 R k )+R k 2 (1 R k 1 )(1 R k ) + R k 3 (1 R k 2 )(1 R k 1 )(1 R k )+... and may be expressed as Z left = R k + i=1 k 1 k R k 1 j=k i+1 (1 R j )) where R i = n i /2 k and n i is the count of rule-table entries belonging to deterministic 2 k i -tuples. 3 A converse procedure gives Z right, and the Z parameter is the greater of Z left and Z right. Examples are given in References 8 and 17. By virtue of being a convergence parameter, Z is also an order-chaos parameter varying from 0 (order) to 1 (chaos). Z can be compared with Langton s well-known parameter [11]. is an order-chaos parameter for CA that may have value greater than binary and measures the density of nonquiescent outputs in a rule-table, so for just binary CA, = c/2 k where c is the count of 1s in a rule-table on k inputs. varies between 0 (order) to 0.5 (chaos) to 1 (order). To allow Z and to be compared, a normalized version of binary is defined [8], ratio =2 c min /2 k where c min is the count of 0s or 1s in the rule-table, whichever is less. ratio must be Z and varies from 0 (order) to 1 (chaos), just as Z. Plots of the G-density against both ratio and Z for the 256 k = 7 totalistic rules, 4 showing the discrepancies as well as similarities, are shown in Figure 9. Points plotted in the top right corner of the ratio graph represent ratio values that do not correspond to behavior as expected. 7. GLIDERS IN 1D CA A large body of literature is devoted the study of space-time patterns in CA. Glider or particle dynamics, where coherent configurations emerge and interact, provide a striking example of self-organization in a simple system. Glider dynamics, and the rules that produce them, have been characterized as complex, in contrast to ordered or chaotic, by Wolfram [19]; that is, those rules yielding localized propagating structures interacting within long transients, where the interactions are clearly interesting. Perhaps the most dramatic example is Conway s 2D game-of-life [9], from where the term glider is borrowed. The human mind is extremely adept at recognizing patterns and identifying those that seem complex and interesting, but it would be extremely useful to have measures that corresponded closely to our subjective classification. An entropy variance measure on the dynamics seems to achieve this, allowing an unlimited source of complex rules to be found, and is further able to characterize rule-space relative to our subjective notions of order, complexity, and chaos. Complex rules are supposed to be rare [10]. Most rules are either ordered or chaotic (see Figure 13), though ordered rules become increasingly rare for larger k. In k = 3 rule-space, the only two sets of glider rules that occur, rule 54 and 110, and their equivalents [8] (see Figures 4 and 14) have been the focus of particular study (e.g., [12]). How a rule is placed within a notional order-complexitychaos space has depended largely on our subjective judgment of typical emergent space-time patterns. Each CA rule self-organizes its patterns in a characteristic way, and these are often recognizable, especially for small k, where a characteristic pattern is apparent even when chaotic, becoming less obvious for larger k. The characteristic pattern of different rules can be analyzed in formal language theory as a regular language with a vocabulary made up of bit sequences and a grammar made up of succession rules between sequences [10], and by a related computational mechanics approach [12]. Glider dynamics correspond to Wolfram s complex class 4 behavior in his classification of CA dynamics [16]. Wolfram orders his classes according to a specific notion of complexity, by the increasing complexity of typical spacetime patterns as measured in formal language theory, and draws analogies with classical continuous dynamical sys- 54 COMPLEXITY 1999 John Wiley & Sons, Inc.

9 FIGURE 9 G -density against both ratio and Z forthesetofk=7totalistic rules, n = 16, for Z The complete basin of attraction field was generated for each rule and garden-of-eden states counted. tems in terms of the attractors typical of each class. His classes are as follows: Class CA dynamics evolves toward A spatially homogeneous state A sequence of simple stable or periodic structures Chaotic aperiodic behavior Complicated localized structures, some propagating... Dynamical systems analogue Limit points Limit cycles Chaotic (strange) attractors Attractors unspecified Langton [11] and others have argued that Wolfram s class 4 more naturally belongs between classes 2 and 3, at a phase transition between order and chaos. Moreover, many ordered rules have both limit points and short limit cycles, though one or the other may predominate, suggesting that class 1 and 2 may usefully be combined. For these reasons, the classification is readjusted as follows: ordered (class 1 2) complex (class 4) chaotic (class 3) What are the essential features of glider behavior? Glider dynamics occur if a limited set of gliders emerge from random initial states and if the interactions between gliders persist for an extended time, which requires that at least some glider collisions create new gliders. Gliders are embedded in a uniform or periodic space-time background or domain, which of necessity has simultaneously emerged. Such a regular domain may be simple (for example, a checkerboard) or a have a more complicated pattern (see Figures 10 12). Distinct chaotic domains may also occur, which cannot support geometrically regular gliders but may support domain walls or particles [20], analogous to gliders. These arise from defects within a chaotic domain (see Figure 16), a boundary between two distinct chaotic domains, or between a chaotic and a regular domain. Domains may be filtered as described in section 9 to show up gliders and domain walls more clearly. Gliders may be regarded as solitary waves within a regular domain and may have the special property of solitons [21], preserving their shape and velocity after interacting with other solitons. Glider velocity varies from 0 to a maximum speed of light of r cells per time-step toward the left or right, where r is the number of cells on the left or right of the target cell. A glider configuration that repeats at each time-step (i.e., with period one) is limited to velocities of 0,1,2,..., r per time-step. Gliders with greater periods may have intermediate fractional velocities. A glider s attributes are the regular domain pattern and spatiotemporal period 1999 John Wiley & Sons, Inc. COMPLEXITY 55

10 FIGURE 10 Interacting gliders with various velocities and backgrounds. 127 time-steps. The k = 5 rule numbers are shown in hex. FIGURE 11 Examples of glider-guns. 127 time-steps. The k = 5 rule numbers are shown in hex. (on both sides of the glider), the glider s temporal period and velocity, its changing diameter, and the list of its repeating configurations. The same description might be applied recursively to each subglider component of a compound glider. Collisions between two glider types often result in a third glider type (or more). One or both gliders may survive a collision with a possible shift in trajectory, or both gliders may be destroyed. A collision may create a temporary chaotic phase before new gliders emerge. The outcome of a collision is sensitive to the point of impact. A glider is often a dislocation or defect of varying width in a domain, which is out of phase on either side of the glider, analogous to a fracture plane in a crystal lattice. Alternatively, a glider may be seen as the zone that reconciles two out-of-phase domains. A glider may separate two entirely different domains, acting as the boundary, as in Figure 11(d). Gliders that eject a stream of subgliders at regular intervals, as in Figure 11, and gliders that survive by absorbing a regular glider stream, as in Figure 10(d), are relatively common. They are analogous to glider guns and eaters, some of the components of the game-of-life universal computer [9]. Because a regular glider stream is essentially the same as a regular domain, a glider gun creates a domain, and an eater absorbs it, so glider guns/eaters are equivalent to a glider forming the boundary between two domains. 56 COMPLEXITY 1999 John Wiley & Sons, Inc.

11 FIGURE 12 (a) A compound glider. (b) A glider with a period of 106 time-steps. (c) A compound glider gun. 168 time-steps. The k = 5 rule numbers are shown in hex. Both the period and diameter of a glider may be considerable. The diameter may show a large variation within the period. Clearly gliders can only emerge in systems large enough to contain them, so that the samples described in section 11 based on n = 150 are biased toward finding relatively small diameter gliders. The existence of compound gliders made up of subgliders colliding periodically may be expected in large enough systems. The example in Figure 12(a) shows a compound glider made from two independent gliders locked in a cycle of repeating collisions. Compound gliders could combine into yet higher-order structures [3], and the process could unfold hierarchically without limit. Once gliders have emerged, CA dynamics could, in principle, be described at a higher level, by glider collision rules as opposed to the underlying CA rules. 8. INPUT-ENTROPY Keeping track of the frequency of rule-table look-ups (the k-block frequency, or look-up frequency ) in a window of time-steps provides a measure, the variance of input-entropy over time, which is used to classify 1D CA automatically for a spectrum of ordered, complex, and chaotic dynamics [22]. The look-up frequency can be represented by a histogram (Figure 13) that distributes the total of n w look-ups amont the 2 k neighborhoods (shown as the fraction of total look-ups), where n = system size, w = the window of timesteps defined, and k = neighborhood size. The Shannon entropy of this frequency distribution, the input-entropy S, at time-step t, for one time-step (w = 1), is given by S t = 2k i=1 (Q t i / n log (Q t i / n)), where Q i t is the look-up frequency of neighborhood i at time t. In practice, the measures are smoothed by being taken over a moving window of time-steps (w = 10 in Figure 13). Figure 13 shows typical examples of ordered, complex, and chaotic dynamics in 1D CA, with input-entropy plots and a snapshot of the look-up frequency histogram alongside. In a random initial state, the different k-blocks occur with equal probability. The start entropy will be correspondingly high. The typical evolution of the input frequency histogram and input-entropy for ordered, chaotic, and complex dynamics is described below. Ordered Dynamics In ordered dynamics, the look-up frequency histogram rapidly becomes highly unbalanced, with most neighborhoods never looked at (their look-up frequency is 0). The few remaining high frequencies settle at constant or periodic values. The entropy will settle at a low constant or periodic value, corresponding to a fixed point or short cycle attractor. Ordered behavior produces extremely short and bushy transient trees with a high density of garden-of-eden states (G-density). Ordered rules decrease disorder and entropy. Complex Dynamics In complex dynamics, the look-up frequency histogram becomes unbalanced, with large erratic fluctuations reflected in the entropy curve. As in ordered behavior, a proportion of neighborhoods are never looked at again after the initial 1999 John Wiley & Sons, Inc. COMPLEXITY 57

12 FIGURE 13 Typical 1D CA space-time patterns showing ordered, complex, and chaotic dynamics (n = 150, k = 5). Alongside each space-time pattern is a plot of the input-entropy (center column) and a snapshot of the look-up frequency histogram averaged over the last 10 time-steps. Only complex dynamics (center row) exhibits high input-entropy variance. sorting-out phase. After an extended time, the system generally settles onto a short attractor cycle. The high frequency neighborhoods correspond to the emergent domain(s); the low frequency neighborhoods, to the interacting gliders. Glider dynamics is subject to two countervailing tendencies: on the one hand, there is a tendency toward order caused by the predominance of domains. But the domains are mobile; their boundaries form the gliders. When these collide, there is a tendency toward chaos. In systems of the size considered, order or chaos may predominate at different times causing the entropy to vary. For large networks where colliding and noncolliding zones coexist, the entropy variance will be reduced to zero in the limit of infinite size. A measure of the variability of the input-entropy curve is its variance or standard deviation. 5 High entropy variance for a significant number of time-steps implies complex space-time dynamics. This includes not just glider dynamics, but also the less frequent dynamics involving domain walls in chaotic domains described earlier in this section. Chaotic Dynamics In chaotic dynamics, the look-up frequency histogram will fluctuate irregularly within a narrow band of low values, and the entropy will fluctuate irregularly within a narrow high band, corresponding to dynamics on very long transients or cycles, analogous to strange attractors in continuous dynamical systems. Transient trees will be sparsely branched, thus will tend to be very long with relatively low G-density. Chaotic rules increase or conserve disorder and entropy. 58 COMPLEXITY 1999 John Wiley & Sons, Inc.

13 FIGURE 14 Examples of filtering space-time patterns to show up gliders more clearly. Left and center, space-time patterns of the k = 3 rule 110, transformed to the equivalent [8] k = 5 rule 3cfc3cfc, n = 150, from the same evolved initial state. Left, cells by value. Center, cells by neighborhood look-up and filtered. Right, space-time patterns of the k = 5 rule 360a96f9 from a random initial state. Cells are shown by neighborhood look-up and are progressively filtered in two stages. About 200 time-steps are shown in each case. 9. FILTERING Current methods for filtering domains in CA space-time patterns are based on a computational mechanics approach [12,20]. An alternative is a byproduct of keeping track of the look-up frequency described in section 8. The frequencies of rule-table look-ups in a moving window of time-steps are recorded. They are also displayed as a changing histogram (Figures 13, 15). The size of the window is 10 for the examples in Figures 14 and 15. To filter background domains, successive key presses in DDLab will progressively suppress the printing of cells that updated with reference to the currently most frequent unsuppressed neighborhood. A dot is shown alongside the look-up frequency histogram indicating which neighborhoods are currently suppressed. The routine can be continued until all neighborhoods are filtered, and reversed to progressively unfilter. Particular neighborhoods can be filtered in isolation. Filtering can be done on-the-fly in DDLab for any rule, including 2D and 3D CA [15]. For most glider rules, only a few neighbourhoods need to be suppressed to filter domains. Rules with very complicated domains, such as the k = 3 rules 54 and 110, must first be transformed to equivalent rules [8] with greater k (k =5 in this case) for successful filtering, which requires suppressing a number of the k = 5 neighborhoods (see Figures 4, 14, and 15). Discontinuities may occur within chaotic domains that nevertheless have regularities in their pattern basis [20], as in the k = 3 rule 18 (see Figure 16), or between two distinct chaotic domains, or between chaotic and regular domains. These types of domains can also be filtered to uncover the domain walls or particles, analogous to gliders. FIGURE 15 Look-up frequency histograms relating to Figure 14. Above, k = 5 rule 360a96f9. Below, k = 3 rule 110 transformed to k = 5 rule 3cfc3cfc. Suppressed neighbourhoods are indicated with a dot John Wiley & Sons, Inc. COMPLEXITY 59

14 FIGURE 16 Unfiltered and partly filtered space-time patterns of k = 3 rule 18 (transformed to k = 5 rule 030c030c). n = 150, about 130 time-steps from the same random initial state, showing discontinuities within the chaotic domain. 10. ENTROPY-DENSITY SIGNATURES A related method of visualizing the entropy variance is to plot input-entropy against the density of 1s relative to a moving window of time-steps. Superimposed plots for a number of complex rules are shown in Figure 17. Each rule produces a characteristic cloud of points that lie within a parabolic envelope because high entropy is most probable at medium density, and low entropy at either low or high density. Each complex rule produces a plot with its own distinctive signature, with high inputentropy variance. Chaotic rules, on the other hand, give a compact cloud at high entropy (at the top of the parabola). For ordered rules, the entropy rapidly falls off with very few data points because the system moves rapidly to an attractor. Gutowitz [23] has also shown entropy-density plots for large samples of rule-space, but his plots show a single point for each rule where the measures on that rule have settled down, whereas the plots shown here focus on the transient history of the system. These plots distinguish order, complexity, and chaos by the vertical extent and density of the cloud. 11. AUTOMATICALLY CLASSIFYING RULE SPACE To distinguish ordered, complex, and chaotic rules automatically, the mean input-entropy taken over a span of time-steps is plotted against the standard deviation of the input-entropy. Figures 18 and 19 FIGURE 17 Entropy-density scatter plots. Input-entropy is plotted against the density of 1s relative to a moving window of time-steps w = 10. k =5,n = 150. Plots for a number of complex rules from the automatic sample (section 11) are show superimposed, each of which has its own distinctive signature, with a marked vertical extent (i.e., high input-entropy variance). About 1,000 time-steps are plotted from several random initial states for each rule. 60 COMPLEXITY 1999 John Wiley & Sons, Inc.

15 FIGURE 18 Classifying a random sample of k = 5 rules by plotting mean entropy against the standard deviation of the entropy. Any standard deviation above the maximum scale has been rescaled to the maximum of summarize how random samples of k = 5, 6 an 7 rules where classified by this method. For each rule, the data were gathered from 5 runs from random initial states, for 430 timesteps, discounting the first 30 to allow the system to settle, with w = 5 as the size of the moving window of time-steps. The measures were averaged, and a point was plotted of mean input-entropy against the standard deviation of the entropy as shown in Figure 18 for the k = 5 sample. To see the frequency distribution of rules, the plots in Figure 19 include an extra axis, making a 2D histogram, representing the number of rules falling within blocks on a grid overlaid over the scatter plot. Looking at the k = 5 2D histogram, the tower in the upper left represents chaotic rules with low standard deviation and high mean entropy. The ridge on the left represents ordered rules with low standard deviation and a spread of lower mean entropy. Complex rules have higher standard deviation and are spread out toward the right. There is a low diagonal valley between the tower and the ridge representing a distinct boundary between ordered and chaotic rules, but a gradual transition from both toward the complex rules. As the standard deviation decreases glider interactions become either more frequent, transients longer, tending toward chaos, or less frequent, transients shorter, tending toward order. The k = 6 and k = 7 plots show an increasing frequency of chaotic rules and a declining frequency of ordered and complex rules as k increases. The decrease in ordered rules is especially marked. The rule samples and measures, including each rule s and Z parameters, were sorted by decreasing standard deviation and decreasing mean entropy for each measure of standard deviation and saved to file. Examples of complex rules from the samples are shown in Figure 20. More examples are presented in Reference 17, and are all available with the DDLab software [14]. Figure 21 shows the same k = 5 rules sample plotting the Z -parameter against standard deviation. To check whether the expected dynamics (recognized subjectively) correspond to the measures as plotted, the dynamics of particular rules at different positions on the plots may be examined very efficiently in DDLab, for example, with a mouse click on the plots in Figures 18 or 21. For the mean entropy-standard deviation plot (Figure 18), preliminary scans indicate that the expected behavior is indeed found, but further investigation is required to properly demarcate the space between ordered, complex, and chaotic rules and to estimate the proportion of different rule classes for different k. For the Z parameter-standard deviation plot (Figure 21), there is an approximate correlation between low Z and order, and high Z and chaos, especially at the extremes. At medium Z, between about 0.5 and 0.75, where most randomly selected rules, and also complex rules, tend to occur, the correlation becomes weaker, Z distinguishes between at least the extremes of order and chaos, and sets a band outside which complex dynamics becomes increasingly unlikely. The automatic samples were generated by DDLab [14]. This may be done for larger system size n and neighborhood k, and the various other parameters may be adjusted. These local measures may be compared to global measures on convergence in attractor basins described in section ATTRACTOR BASIN MEASURES Measures on attractor basins include the number of attractors, attractor periods, size of basins, characteristic length of 1999 John Wiley & Sons, Inc. COMPLEXITY 61

16 FIGURE 19 Left, classifying a random sample of k =5 rules by plotting mean entropy against standard deviation of the entropy, with the frequency of rules within a grid shown vertically. Below, equivalent plots for samples of k = 6 and 7 rules. transients, and the characteristic branching within trees. The last in particular gives a good measure of the convergence of the dynamical flow in state-space. The simplest measure that captures the degree of convergence is the density of garden-of-eden state [24], G-density, counted in attractor basins or subtrees, and its rate of increase with n as shown in Figure 22. A more comprehensive measure is the in-degree frequency distribution, plotted as a histogram. The in-degree of a state is the number of its immediate pre-images. This can be taken on attractor basins, on just a subtree, or on part of a subtree for larger systems. Subtrees are portrayed as graphs showing trajectories merging onto the subtree root state. Examples of in-degree histograms for a typical subtree for ordered, complex, and chaotic rules are shown in Figure 23. The horizontal axis represents in-degree size, from zero (garden-of-eden states) upward, the vertical axis represents the frequency of the different in-degrees. The system size n = 50 for the complex and chaotic rules. For very ordered rules, in-degrees become astronomical. The ordered rule shown is only moderately ordered; however, the system size was reduced to n = 40 to allow easier computation. From the preliminary data gathered so far, the profile of the in-degree histogram for different classes of rule is as follows: Ordered: Very high garden-of-eden frequency and significant frequency of high in-degrees. High convergence. Complex: Approximates a power law distribution. Medium convergence. Chaotic: Lower garden-of-eden frequency compared to complex rules, and a higher frequency of low indegrees. Low convergence. Issues for further investigation are: a systematic look at the in-degree histogram profiles relative to rules at various positions on the mean entropy/standard deviation scatter plots, how profiles change with system size, if a subtree fragment is representative of the dynamics as a whole, if the 62 COMPLEXITY 1999 John Wiley & Sons, Inc.

17 FIGURE 20 Examples of k = 5, 6, and 7 complex space-time patterns, with high standard deviation, from the automatic samples. n = 150, 150 time-steps from random initial states. Cells are colored according to the neighborhood. profile changes for subtrees deep in a basin of attraction as opposed to close to the outer leaves, and to look at the part of subtrees close to particular trajectories (within a given distance in reverse time-steps), especially in relation to glider dynamics. 13. GLIDER INTERACTIONS AND BASINS OF ATTRACTION It is possible to identify classes of configurations that make up different components of attractor basins in glider rules. In random states, configurations occur with equal probability, so the special glider/background configurations are unlikely. Nonglider/background states make up the majority of state-space and are likely to be garden-of- Eden states or states just a few steps forward in time from garden-of-eden states. They occur in the initial sorting-out phase of the dynamics and appear as short bushy dead-end side branches along the length of long transients, as well as at their tips. States dominated by glider and background and configurations are special cases, a small subcategory of state-space. They constitute the glider interaction phase, making up the main lines of flow within the long transients. This has also been noted by Domain [25], who described the main lines of flow as the topological skeleton of physically relevant states and the short dead-end side branches from garden-of-eden states as a skin of nonphysical transitional states, comprising the bulk of the states in an attractor basin. Gliders in the interaction phase can be regarded as competing subattractors, with the final survivors persisting in the attractor cycle. Finally, states made up solely of noninteracting gliders (i.e., having equal velocity), or domains free of gliders, must cycle and therefore constitute the relatively short attractors, with a period depending on the glider 1999 John Wiley & Sons, Inc. COMPLEXITY 63

18 FIGURE 21 The same random sample of k = 5 rules as in Figure 18, lef the Z parameter against standard deviation of the entropy and right with a vertical frequency axis as in Figure 19. velocity. The attractor states are made up of gliders, compound gliders, or just domains, and thus form a tiny subcategory of state-space. By simply looking at the space-time patterns of a glider rule from a number of different initial states, most gliders in its glider repertoire (relative to the system size) may be identified. A complete list would allow a description of most of the attractors by finding all possible permutation of noninteracting gliders. 14. DISCUSSION Complex behavior in 1D CA, especially the emergence of gliders, mirrors our intuitive notion of complex forms and processes emerging in nature. These are arguably the simplest systems where complex phenomena arise. Their simplicity allows a description of global as well as local behavior and how this varies across rule-space. A global perspective on CA dynamics and rule-space is provided by the notion of attractor basins. The basin of attraction fields of complex rules are typically composed of moderately bushy transient trees rooted on relatively short attractor cycles. Gliders interacting aperiodically belong to the main lines of flow within the transient trees. Configurations where gliders interact periodically, or have ceased to interact, make up the attractor cycles. Gliders have a distinct identity. Their interactions are predictable. A collision table could be formulated empirically, without knowing the underlying rule-table mechanism. The collision table would probably need to hold much more information than the rule-table. It would need to describe all possible collisions at different points of impact between gliders. However, compared to the rule-table, the collision table would provide a far more useful description of established behavior, enabling some prediction of the system s future evolution, though only the rule-table could account for the origins of gliders, their emergence by a process of self-organization from random patterns. Interacting gliders may combine to create compound gliders, interacting at yet higher levels of description, and conceivably the process could unfold hierarchically without limit in large enough systems. This is analogous to describing matter in terms of chemistry as opposed to the under- FIGURE 22 The G-density plotted against system size n for the ordered, complex, and chaotic rules shown in Figures 13 and 23. The entire basin of attraction field was generated for n = 7 to 22, and gardenof-eden states counted. The relative G-density and rate of increase with n provides a simple measure of convergence. 64 COMPLEXITY 1999 John Wiley & Sons, Inc.

19 FIGURE 23 Ordered dynamics. Rule 01dc3610, n = 40, Z = , ratio = Right: The complete subtree 7 levels deep, with nodes, G-density = Complex dynamics. Rule 6c1e53a8, n = 50, Z = 0.727, ratio = Right: The subtree, stopped after 12 levels, with nodes, G-density = Chaotic dynamics. Rule 994a6a65, n = 50, Z = 0.938, ratio = Right: The subtree, stopped after about 75 levels, with 9446 nodes, G-density = Ordered complex chaotic CA dynamics. The space-time patterns of the rules are shown in Figure 13. The in-degree histogram of a typical subtree shown in normal and log-log form. lying subatomic particles, or in terms of biology as opposed to the underlying chemistry. There are any number of further analogies that might be drawn from nature or society. However, the origins of the higher-level entities must refer to the lower level. According to this approach, a system s complexity is the number of levels of description that underpin it in a descending hierarchy. An unlimited source of complex rules that support gliders is available by the automatic method described, based on local measures, in particular input-entropy variance, which also classifies rule-space for a spectrum of ordered, complex, and chaotic dynamics. Global measures, G- density, and in-degree frequency, taken on attractor basins and subtrees, relate to the local measures. Both local and global measures relate approximately to the rule parameter Z. Further systematic investigations of both the local and global measures, based on the automatic rule samples, and extended samples are needed for a deeper understanding of CA rule-spaces. The computer tools for such an investigation are largely in place John Wiley & Sons, Inc. COMPLEXITY 65

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